arXiv:1501.07421v4 [math-ph] 16 Jun 2016
BETHE ANSATZ AND THE SPECTRAL THEORY OF AFFINE LIE ALGEBRA–VALUED CONNECTIONS.
THE SIMPLY–LACED CASE
DAVIDE MASOERO, ANDREA RAIMONDO, DANIELE VALERI
Abstract. We study the ODE/IM correspondence for ODE associated to b
g-valued connections, for a simply-laced Lie algebra g. We prove that subdom-inant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called Ψ-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
Contents
0. Introduction 1
1. Affine Kac-Moody algebras and their finite dimensional representations 4
2. Asymptotic Analysis 9
3. The Ψ-system 14
4. The Q-system 18
5. g-Airy function 22
Appendix A. Action of Λ in the fundamental representations 24
References 27
0. Introduction
The ODE/IM correspondence, developed since the seminal papers [13, 3], con-cerns the relations between the generalized spectral problems of linear Ordinary Differential Equations (ODE) and two dimensional Integrable Quantum Field The-ories (QFT), or continuous limit of Quantum Integrable Systems. Accordingly, the acronym IM usually refers to Integrable Models (or Integrals of Motion). More re-cently, it was observed that the correspondence can also be interpreted as a relation between Classical and Quantum Integrable Systems [27], or as an instance of the Langlands duality [17].
The original ODE/IM correspondence states that the spectral determinant of certain Schrödinger operators coincides with the vacuum eigenvalue Q(E) of the Q-operators of the quantum KdV equation [4], when E is interpreted as a parameter of the theory. This correspondence has been proved [13, 3] by showing that the spectral determinant and the eigenvalues of the Q operator solve the same functional equation, namely the Bethe Ansatz, and they have the same analytic properties.
The correspondence has soon been generalized to higher eigenvalues of the Quan-tum KdV equation – obtained modifying the potential of the Schrödinger operator [5] – as well as to Conformal Field Theory with extended Wn-symmetries, by
consid-ering ODE naturally associated with the simple Lie algebra An(scalar linear ODEs
of order n+ 1) [10,34]. Within this approach, the original construction corresponds to the Lie algebra A1, and the associated ODE is the Schrödinger equation.
After these results, it became clear that a more general picture including all simple Lie algebras – as well as massive (i.e. non conformal) Integrable QFT – was missing. While the latter issue was addressed in [20,27] (and more recently in [11,1,30]), the ODE/IM correspondence for simple Lie algebras different from An
has not yet been fully understood.
The first work proposing an extension of the ODE/IM correspondence to clas-sical Lie algebras other than An is [9], where the authors introduce a g-related
scalar ODE as well as the important concept of Ψ-system. The latter is a set of quadratic relations that solutions of the g-related ODE are expected to satisfy in different representations. It was then recognized in [17] that the g-related scalar differential operators studied in [5,9] can be regarded as affine opers associated to the Langlands dual of the affine Lie algebra bg. This discovery gave the theory a solid algebraic footing – based on Kac-Moody algebras and Drinfeld-Sokolov Lax operators – which was further investigated in [32] and that we hereby follow.
In the present paper we establish the ODE/IM correspondence for a simply laced Lie algebra g, in which case the affine Lie algebrabg coincides with its Langlands dual [17]. More concretely, we prove that for any simple Lie algebra g of ADE type there exists a family of spectral problems, which are encoded by entire functions Q(i)(E),
i = 1, . . . , n = rank g, satisfying the ADE-Bethe Ansatz equations [31,35,2]:
n Y j=1 ΩβjCij Q(j)ΩCij2 E∗ Q(j)Ω−Cij2 E∗ = −1 , (0.1)
for every E∗ ∈ C such that Q(i)(E∗) = 0. In equation (0.1), C = (C
ij)ni,j=1 is
the Cartan matrix of the algebra g, while the phase Ω and the numbers βj are the
free-parameters of the equation. More precisely, following [17] we introduce the bg-valued connection
L(x, E) = ∂x+
ℓ
x+ e + p(x, E)e0, (0.2)
where ℓ ∈ h is a generic element of the Cartan subalgebra h of g, e0, e1, . . . , enare the
positive Chevalley generators of the affine Kac-Moody algebrabg and e =Pni=1ei.
In addition, the potential1
is p(x, E) = xM h∨
− E, where M > 0 and h∨is the dual
Coxeter number of g.
After [32], for every fundamental representation 2
V(i) of bg we consider the
differential equation
L(x, E)Ψ(x, E) = 0 . (0.3)
The above equation has two singular points, namely a regular singularity in x = 0 and an irregular singularity in x = ∞, and a natural connection problem arises when one tries to understand the behavior of the solutions globally. We show that for every representation V(i) there exists a unique solution Ψ(i)(x, E) to equation
(0.3) which is subdominant for x → +∞ (by subdominant we mean the solution that goes to zero the most rapidly). Then, we define the generalized spectral determinant Q(i)(E; ℓ) as the coefficient of the most singular (yet algebraic) term
of the expansion of Ψ(i)(x, E) around x = 0. Finally, we prove that for generic ℓ ∈ h
the generalized spectral determinants Q(i)(E; ℓ) are entire functions and satisfy the
Bethe Ansatz equation (0.1), also known as Q-system.
The paper is organized as follows. In Section 1 we review some basic facts about the theory of finite dimensional Lie algebras, affine Kac-Moody algebras and
1
We stick to a potential of this form for simplicity. However all proofs work with minor modifications for a more general potential discussed in [5,17].
2
Actually V(i)is an evaluation representation of the i-th fundamental representation of g.
their finite dimensional representations. Furthermore, we introduce the relevant representations that we will consider throughout the paper.
Section 2 is devoted to the asymptotic analysis of equation (0.3). The main result is provided by Theorem2.4, from which the existence of the fundamental (subdom-inant) solution Ψ(i)(x, E) follows. The asymptotic properties of the solution turn
out to depend on the spectrum of Λ = e0+ e ∈bg, as observed in [32].
The main result of Section 3 is Theorem 3.6, which establishes the following quadratic relation among the functions Ψ(i)(x, E), known as Ψ-system, that was
conjectured in [9]: mi Ψ(i)−1 2 (x, E) ∧ Ψ(i)1 2 (x, E)= ⊗j∈IΨ(j)(x, E)⊗Bij. (0.4) Here, Ψ(i)±1 2
(x, E) is the twisting of the solution Ψ(i)(x, E) defined in (2.8), while
mi is the morphism ofbg-modules defined by (1.11), and B = (Bij)ni,j=1 denotes
the incidence matrix of g. In order to prove Theorem 3.6, we first establish in Proposition3.4 some important properties of the eigenvalues of Λ on the relevant representations previously introduced. More precisely, we show that in each repre-sentation V(i) there exists a maximal positive eigenvalue λ(i) (see Definition2.3),
and that the following remarkable identity holds: e−hπi∨ + e πi h∨ λ(i) = n X j=1 Bijλ(j), i = 1, . . . , n = rank g . (0.5)
Equation (0.5) shows that the vector (λ(1), . . . , λ(n)) is the Perron-Frobenius
eigen-vector of the incidence matrix B, and this implies that the λ(i)’s are (proportional
to) the masses of the Affine Toda Field Theory with the same underlying Lie algebra g[18].
In Section4 we derive the Bethe Ansatz equations (0.1). To this aim, we study the local behavior of equations (0.3) close to the Fuchsian singularity x = 0, and we define the generalized spectral determinants Q(i)(E; ℓ) and eQ(i)(E; ℓ) using some
properties of the Weyl group of g. In addition, using the Ψ-system (0.4) we prove Theorem 4.3, which gives a set of quadratic relations among the generalized spec-tral determinants (equation (4.6)). Evaluating these relations at the zeros of the functions Q(i)(E; ℓ), we obtain the Bethe Ansatz equations (0.1).
Finally, in Section 5, we briefly study an integral representation of the sub-dominant solution of equation (0.3) with a linear potential p(x, E) = x, and we compute the spectral determinants Q(i)(E; ℓ). Some explicit computations of the
eigenvectors of Λ are provided in Appendix A.
In this work we do not study the asymptotic behavior of the Q(i)(E)’s, for E ≪ 0,
that was conjectured in [9], neither we address the problem of solving the Bethe Ansatz equations (0.1) – for a given asymptotic behavior of the functions Q(i)(E)’s
– via the Destri-de Vega integral equation [8]. These problems are of a rather different mathematical nature with respect to the ones treated in this work, and they will be considered in a separate paper.
Note added in press. We proved the ODE/IM correspondence for non simply– laced Lie algebras in [29].
Acknowledgments. The authors were partially supported by the INdAM-GNFM “Progetto Giovani 2014”. D. M. is supported by the FCT scholarship, number SFRH/BPD/75908/2011. D.V. is supported by an NSFC “Research Fund for In-ternational Young Scientists” grant. Part of the work was completed during the visit of A.R. and D.V. to the Grupo de Física Matemática of the Universidade de Lisboa, and during the participation of the authors to the intensive research program “Perspectives in Lie Theory” at the Centro di Ricerca Matematica E. De Giorgi in Pisa. We wish to thank these institutions for their kind hospitality. D.M thanks the Dipartimento di Fisica of the Università degli Studi di Torino for the kind hospitality. A. R. thanks the Dipartimento di Matematica of the Università degli Studi di Genova for the kind hospitality. We also wish to thank Vladimir Bazhanov, Alberto De Sole, Edward Frenkel, Victor Kac and Roberto Tateo for useful discussions.
While we were writing the present paper, we became aware of the work [12] (see [30] for a more detailed exposition) that was being written by Patrick Dorey, Ste-fano Negro and Roberto Tateo. Their paper focuses on ODE /IM correspondence for massive models related to affine Lie algebras and deals with similar problems in representation theory. We thank Patrick, Stefano and Roberto for sharing a preliminary version of their paper.
1. Affine Kac-Moody algebras and their finite dimensional representations
In this section we review some basic facts about simple Lie algebras, affine Kac-Moody algebras and their finite dimensional representations which will be used throughout the paper. The discussion is restricted to simple Lie algebras of ADE type, but most of the results hold also in the non-simply laced case; we refer to [19,24] for further details. At the end of the section we introduce a class ofbg-valued connections and the associated differential equations.
1.1. Simple lie algebras and fundamental representations. Let g be a simple finite dimensional Lie algebra of ADE type, and let n be its rank. The Dynkin diagrams associated to these algebras are given in Table 1. Let C = (Cij)i,j∈I
be the Cartan matrix of g, and let us denote by B = 21n− C the corresponding incidence matrix. Since g is simply-laced, it follows that Bij = 1 if the nodes i, j
of the Dynkin diagram of g are linked by an edge, and Bij= 0 otherwise.
An α1 α2 . . . αn−1 αn E6 α1 α2 α3 α5 α6 α4 Dn α1 α2 . . . αn−2 αn αn−1 E7 α1 α2 α3 α4 α6 α7 α5 E8 α1 α2 α3 α4 α5 α7 α8 α6
Table 1: Dynkin diagrams of simple Lie algebras of ADE type.
Let {fi, hi, ei | i ∈ I = {1, . . . , n}} ⊂ g be the set of Chevalley generators of g.
They satisfy the following relations (i, j ∈ I):
[hi, hj] = 0 , [hi, ej] = Cijej, [hi, fj] = −Cijfj, [ei, fj] = δijhi, (1.1)
together with the Serre’s identities. Recall that
h=M
i∈I
Chi⊂ g
is a Cartan subalgebra of g with corresponding Cartan decomposition
g= h ⊕ M
α∈R
gα
!
. (1.2)
As usual, the gαare the eigenspaces of the adjoint representation and R ⊂ h∗is the
set of roots. For every i ∈ I, let αi ∈ h∗be defined by αi(hj) = Cij, for every j ∈ I.
Then gαi = Ceαi. Hence, ∆ = {αi | i ∈ I} ⊂ R. The elements α1, . . . , αn are
called the simple roots of g. They can be associated to each vertex of the Dynkin diagram as in Table1. We denote by
P = {λ ∈ h∗| λ(hi) ∈ Z, ∀i ∈ I} ⊂ h∗
the set of weights of g and by
P+ = {λ ∈ P | λ(hi) ≥ 0, ∀i ∈ I} ⊂ P
the subset of dominant weights of g. We recall that we can define a partial ordering on P as follows: for λ, µ ∈ P , we say that λ < µ if λ − µ is a sum of positive roots. For every λ ∈ h∗ there exists a unique irreducible representation L(λ) of g such
that there is v ∈ L(λ) \ {0} satisfying
hiv = λ(hi)v , eiv = 0 , for every i ∈ I .
L(λ) is called the highest weight representation of weight λ, and v is the highest weight vector of the representation. We have that L(λ) is finite dimensional if and only if λ ∈ P+, and conversely, any irreducible finite dimensional representation of
gis of the form L(λ) for some λ ∈ P+. For λ ∈ P+, the representation L(λ) can be
decomposed in the direct sum of its (finite dimensional) weight spaces L(λ)µ, with
µ ∈ P , and we have that L(λ)µ6= 0 if and only if µ < λ. We denote by Pλ the set
of weights appearing in the weight space decomposition of L(λ); the multiplicity of µ in the representation L(λ) is defined as the dimension of the weight space L(λ)µ.
Recall that the fundamental weights of g are those elements ωi∈ P+, i ∈ I,
satis-fying
ωi(hj) = δij, for every j ∈ I . (1.3)
The corresponding highest weight representations L(ωi), i ∈ I, are known as
funda-mental representations of g, and for every i ∈ I we denote by vi∈ L(ωi) the highest
weight vector of the representation L(ωi). Since the simple roots and the
funda-mental weights of g are related via the Cartan matrix (which is non-degenerate) by the relation
ωi=
X
j∈I
(C−1)jiαj, i ∈ I ,
then we can associate to the i−th vertex of the Dynkin diagram of g the corre-sponding fundamental representation L(ωi) of g.
1.2. The representations L(ηi). Let us consider the dominant weights
ηi=
X
j∈I
Bijωj, i ∈ I ,
where B = (Bij)i,j∈I is the incidence matrix defined in Section 1.1, and let L(ηi)
be the corresponding irreducible finite dimensional representations. In addition, we consider the tensor product representations
O
j∈I
L(ωj)⊗Bij, i ∈ I .
We now show that we can find a copy of the irreducible representation L(ηi) inside
the representations V2L(ωi) and Nj∈IL(ωj)⊗Bij. This will be used in Section 3
to construct the so-called ψ-system.
Lemma 1.1. The following facts hold in the representationV2L(ωi).
(a) For every j ∈ I, we have that
ej(fivi∧ vi) = 0 .
(b) For every h ∈ h, we have that
h(fivi∧ vi) = ηi(h)(fivi∧ vi) .
(c) Any other weight of the representationV2L(ωi) is smaller than ηi.
Proof. Since vi is a highest weight vector for L(ωi) we have that ejvi = 0 for every
j ∈ I. Moreover, using the relations [ej, fi] = δijhi together with equation (1.3),
we have
ejfivi= fiejvi+ δijhivi= δijvi,
and therefore ej(fivi ∧ vi) = δijvi ∧ vi = 0, proving part (a). By linearity, it
suffices to show part (b) for h = hi, i ∈ I. Recall that, by equation (1.1), we have
[hi, fj] = −Cijfj, for every i, j ∈ I. Then,
hjfivi= (fihj+ [hj, fi]) vi = (δij− Cji)fivi.
Hence, hj(fivi∧ vi) = (2δij − Cji)(fivi∧ vi) = ηi(hj)(fivi∧ vi). Finally, let w 6=
ωi, ηi− ωi be a weight appearing in L(ωi). It follows from representation theory of
simple Lie algebras that
ω < ηi− ωi= ωi−
X
j∈I
Cjiωj< ωi.
Therefore, by part (b), the maximal weight of V2L(ωi) is wi+ ηi− ωi = ηi, thus
proving part (c).
As a consequence of the above Lemma it follows that fivi∧vi∈V2L(ωi) is a highest
weight vector of weight ηi. Since g is simple, the subrepresentation of V2L(ωi)
generated by the highest weight vector fivi ∧ vi is irreducible. Therefore, it is
isomorphic to L(ηi). By the complete reducibility ofV2L(ωi) we can decompose
it as follows:
2
^
L(ωi) = L(ηi) ⊕ U ,
where U is the direct sum of all the irreducible representations different from L(ηi).
By an abuse of notation we denote with the same symbol the representation L(ηi)
and its copy in V2L(ωi). For every i ∈ I, let us denote by wi = ⊗j∈Ivj⊗Bij. The
following analogue of Lemma1.1in the case of the representationNj∈IL(ωj)⊗Bij
holds true.
(a) For every j ∈ I, we have that
ejwi= 0 .
(b) For every h ∈ h, we have that
hwi= ηi(h)wi.
(c) Any other weight of the representationNj∈IL(ωj)⊗Bij is smaller than ηi.
Proof. Same as the proof of Lemma 1.1.
As for the previous discussion, by Lemma1.2it follows that wi∈Nj∈IL(ωj)⊗Bij
is a highest weight vector which generates an irreducible subrepresentation isomor-phic to L(ηi). Lemmas 1.1 and 1.2, together with the Schur Lemma, imply that
there exists a unique morphism of representations
mi: 2 ^ L(ωi) → O j∈I L(ωj)⊗Bij, (1.4)
such that Ker mi= U and mi(fivi∧ vi) = wi.
1.3. Affine Kac-Moody algebras and finite dimensional representations. Let h∨ be the dual Coxeter number of g. Let us denote by κ the Killing form of g
and let us fix the following non-degenerate symmetric invariant bilinear form on g (a, b ∈ g):
(a|b) = 1
h∨κ(a|b) .
Let g ⊗ C[t, t−1] be the space of Laurent polynomials with coefficients in g. For
a(t) =PNi=−Maiti∈ C[t, t−1], we let
Rest=0a(t)dt = a−1.
The affine Kac-Moody algebrabg is the vector space bg = g ⊗ C[t, t−1] ⊕ Cc endowed
with the following Lie algebra structure (a, b ∈ g, f (t), g(t) ∈ C[t, t−1]):
[a ⊗ f (t), b ⊗ g(t)] = [a, b] ⊗ f (t)g(t) + (a|b) Rest=0(f′(t)g(t)dt) c ,
[c,bg] = 0 . (1.5)
The set of Chevalley generators ofbg is obtained by adding to the Chevalley genera-tors of g some new generagenera-tors f0, h0, e0(for the construction, see for example [24]).
The generator e0, which plays an important role in the paper, can be constructed
as follows. There exists a root −θ ∈ R, known as the lowest root of g, such that −θ − αi6∈ R, for every i ∈ I. Then
e0= a ⊗ t , for some a ∈ g−θ. (1.6)
We now consider an important class of representations of bg. Let V be a finite dimensional representation of g. For ζ ∈ C∗ we define a representation ofbg, which
we denote by V (ζ), as follows: as a vector space we take V (ζ) = V , and the action ofbg is defined by
(a ⊗ f (t))v = f (ζ)(av) , c v = 0 , for a ∈ g , f (t) ∈ C[t, t−1] , v ∈ V . The representation V (ζ) is called an evaluation representation of bg. If V and W are representations of g then any morphism of representations f : V → W can obviously be extended to a morphism of representations f : V (ζ) → W (ζ), which we denote by the same letter by an abuse of notation. Similarly, when referring to weights or weight vectors of the evaluation representation V (ζ), we mean the weights and weight vectors of the representation V with respect to the action of g.
For a representation V of g and k ∈ C, we denote Vk= V (e2πik) the evaluation
representation of bg corresponding to ζ = e2πik. Clearly, if k ∈ Z then V
k = V0 =
V (1), and if k ∈ 1
2+ Z then Vk = V12 = V (−1). Then, for each vertex i ∈ I of the
Dynkin diagram of g, we associate the following finite dimensional representation ofbg:
V(i) = L(ωi)p(i)
2 , (1.7)
where L(ωi) is the fundamental representation of g with highest weight ωi, and the
function p : I → Z/2Z is defined inductively as follows:
p(1) = 0 , p(i) = p(j) + 1 , for j < i such that Bij> 0. (1.8)
By an abuse of language we call the representations (1.7), for i ∈ I, the finite dimensional fundamental representations of bg. These representations will play a fundamental role in the present paper.
Example 1.3. In type An, V(1) = L(ω1)0is the evaluation representation at ζ = 1
of the standard representation L(ω1) = Cn+1. Moreover,
V(i)= i ^ Vi−1(1) 2 , for k = 1, . . . , n .
Example 1.4. In type Dn, V(1)= L(ω1)0is the evaluation representation at ζ = 1
of the standard representation L(ω1) = C2n. Moreover,
V(i)= i ^ Vi−1(1) 2 , for k = 1, . . . , n − 2 . Finally, V(n−1)= L(ω n−1)n 2 and V (n)= L(ω n)n
2 are the evaluation representations
at ζ = (−1)n of the so-called half-spin representations.
Example 1.5. In type E6, V(1) = L(ω1)0 and V(5) = L(ω5)0 are the evaluation
at ζ = 1 of the two 27-dimensional representations (they are dual to each other). Moreover, V(2)= 2 ^ V1(1) 2 , V(3)= 3 ^ V0(1)∼= 3 ^ V0(5), V(4)= 2 ^ V1(5) 2 . Finally, V(6) = L(ω6)1
2 is the evaluation representation at ζ = −1 of the adjoint
representation.
As a final remark, note that by equation (1.7) we have
2 ^ L(ωi) ! p(i)+1 2 = 2 ^ V1(i) 2 , (1.9)
and if we introduce the evaluation representation M(i)=O
j∈I
V(j)⊗Bij
, ∀i ∈ I , (1.10)
then by equation (1.7), and from the fact that Bij 6= 0 implies p(i) = p(j) + 1, it
follows that the morphism mi defined by equation (1.4) extends to a morphism of
evaluation representations mi: 2 ^ V1(i) 2 → M (i). (1.11)
The construction of the ψ−system will be provided by means of the above extended morphism. Due to Lemmas1.1and1.2, the evaluation representation
W(i) = L(ηi)p(i)+1
2 , i ∈ I , (1.12)
is a subrepresentation of both V2V1(i)
2 and M
(i), and the copies of W(i) in these
representations are identified by means of the morphism mi.
1.4. bg-valued connections and differential equations. Let {fi, hi, ei | i =
0, . . . , n} ⊂bg be the set of Chevalley generators of bg, and let us denote e =Pni=1ei.
Fix an element ℓ ∈ h. Following [17] (see also [32]), we consider the bg-valued connection
L(x, E) = ∂x+
ℓ
x+ e + p(x, E)e0, (1.13)
where p(x, E) = xM h∨
− E, with M > 0 and E ∈ C. Since e is a principal nilpotent element, it follows from the Jacobson-Morozov Theorem together with equation (1.6) that there exists an element h ∈ h such that
[h, e] = e , [h, e0] = −(h∨− 1)e0, (1.14)
see for instance [7]. Under the isomorphism between h and h∗ provided by the
Killing form, the element h corresponds to the Weyl vector (half sum of positive roots). Let k ∈ C and introduce the quantities
ω = eh∨ (M+1)2πi , Ω = e2πiMM+1 = ωh ∨
M.
Moreover, let Mk be the automorphism ofbg-valued connections which fixes ∂x, g
and c and sends t → e2πikt. Then from equations (1.5), (1.13) and (1.14) we get
Mk ωkad hL(x, E)
= ωkL(ωkx, ΩkE) . (1.15)
We denote Lk(x, E) = Mk(L(x, E)), for every k ∈ C. Note in particular that
L0(x, E) = L(x, E). Equation (1.15) implies that for a given finite dimensional
representation V of g, the action of Lk(x, E) on the evaluation representation V0
ofbg is the same as the action of L(x, E) on Vk, for every k ∈ C. In addition, let bC
be the universal cover of C∗, suppose ϕ(x, E) : bC→ V
0 is a family – depending on
the parameter E – of solutions of the (system of) ODE
L(x, E)ϕ(x, E) = 0 , (1.16)
and for k ∈ C introduce the function
ϕk(x, E) = ω−khϕ(ωkx, ΩkE) . (1.17)
Since on any evaluation representation of bg we have
ead ab = eabe−a, a, b ∈bg , (1.18) then by equations (1.15) and (1.18), we have that
Lk(x, E)ϕk(x, E) = 0 , (1.19)
namely ϕk(x, E) : bC→ Vk is a solution of (1.16) for the representation Vk.
2. Asymptotic Analysis
Let V be an evaluation representation ofbg. Let us denote by eC= C \ R≤0, the complex plane minus the semi-axis of real non-positive numbers, and let us consider a solution Ψ : eC→ V of L(x, E)Ψ(x) = Ψ′(x) + ℓ x+ e + p(x, E)e0 Ψ(x) = 0 . (2.1)
Equation (2.1) is a (system of) linear ODEs, with a Fuchsian singularity at x = 0 and an irregular singularity at x = ∞. Our goal is to prove the existence of solutions of equation (2.1) with a prescribed asymptotic behavior at infinity in a Stokes sector of the complex plane. In the present form, however, equation (2.1) falls outside the reach of classical results such as the Levinson theorem [26] or the
complex Wentzel-Kramers-Brillouin (WKB) method [16], since the most singular term p(x, E)e0 is nilpotent. In order to study the asymptotic behavior of solutions
to (2.1) we use a slight modification of a gauge transform originally introduced in [32], and we then prove the existence of the desired solution for the new ODE by means of a Volterra integral equation.
We now consider the required transformation. First, we note that the function p(x, E)h1∨ has the asymptotic expansion
p(x, E)h1∨ = q(x, E) + O(x−1−δ), where δ = M (h∨(1 + s) − 1) − 1 > 0, s = ⌊M + 1 h∨M ⌋, (2.2) and q(x, E) = xM + s X j=1 cj(E)xM(1−h ∨ j). (2.3)
For every j = 1, . . . , s, the function cj(E) is a monomial of degree j in E.
Lemma 2.1. Let L(x, E) be thebg-valued connection defined in (1.13) and let h ∈ h satisfy relations (1.14). Then, we have the following gauge transformation
q(x, E)ad hL(x, E) = ∂x+ q(x, E)Λ +
ℓ − M h
x + O(x
−1−δ) , (2.4)
where Λ = e0+ e, the function q(x, E) is given by (2.3) and δ by (2.2).
Proof. It follows by a straightforward computation using relations (1.14). Remark 2.2. The transformation (2.4) differs from the one used in [32] and given by xMad hL(x, E) = ∂x+ xMΛ + ℓ − M h x − e0E xM(h∨−1).
The latter transformation fails to be the correct if M (h∨− 1) ≤ 1. In fact, in this
case the term e0E/xM(h
∨
−1)goes to zero slowly and alters the asymptotic behavior.
We are then left to analyze the equation
Ψ′(x) + q(x, E) Λ +ℓ − M h x + A(x) Ψ(x) = 0 , (2.5)
where A(x) = O(x−1−Mh∨
) is a certain matrix-valued function. From the general theory we expect the asymptotic behavior to depend on the primitive of q(x, E), which is known as the action. In the generic case M+1
h∨M ∈ Z/ +, the action is defined
as
S(x, E) = Z x
0
q(y, E)dy , x ∈ eC, (2.6)
where we chose the branch of q(x, E) satisfying q ∼ |x|M for x real. In the case M+1
h∨M ∈ Z+ then formula (2.6) becomes
S(x, E) = s−1 X j=0 Z x 0 cj(E)yM(1−h ∨ j)dy + c s(E) log x , s = M + 1 h∨M .
Existence and uniqueness of solutions to equation (2.1) (or equivalently, of (2.5)) depend on the properties of the element Λ = e+e0in the representation considered.
We therefore introduce the following
Definition 2.3. Let A be an endomorphism of a vector space V . We say that a eigenvalue λ of A is maximal if it is real, its algebraic multiplicity is one, and λ > Re µ for every eigenvalue µ of A.
We are now in the position to state the main result of this section.
Theorem 2.4. Let V be an evaluation representation of bg, and let the matrix representing Λ ∈ bg in V have a maximal eigenvalue λ. Let ψ ∈ V be the corre-sponding unique (up to a constant) eigenvector. Then, there exists a unique solution Ψ(x, E) : eC→ V to equation (2.1) with the following asymptotic behavior:
Ψ(x, E) = e−λS(x,E)q(x, E)−h ψ + o(1) as x → +∞ . Moreover, the same asymptotic behavior holds in the sector | arg x| < π
2(M+1), that
is, for any δ > 0 it satisfies
Ψ(x, E) = e−λS(x,E)q(x, E)−h ψ + o(1), in the sector | arg x| < π
2(M + 1)− δ . (2.7) The function Ψ(x, E) is an entire function of E.
Remark 2.5. We will prove in Section 3 that the hypothesis of the theorem are satisfied in the ADE case for the representations V(i), V2
V1(i) 2
, M(i) and W(i),
which were introduced in the previous section.
Remark 2.6. In the case M (h∨− 1) > 1, the asymptotic expansion of the solution
Ψ in the above theorem reduces to
Ψ(x) = e−λxM +1M+1q(x, E)−h ψ + o(1), in the sector | arg x| < π
2(M + 1). Before proving Theorem 2.4, we apply it to prove the existence of linearly inde-pendent solutions to equation (2.1). First, note that for any k ∈ R such that |k| < h∨(M+1)2 , the function
Ψk(x, E) = ω−khΨ(ωkx, ΩkE) , x ∈ R+ (2.8)
defines, by analytic continuation, a solution Ψk : eC→ Vk of equation (2.1) in the
representation Vk. Theorem2.4implies the following result.
Corollary 2.7. For any k ∈ R such that |k| < h∨(M+1)2 , on the positive real semi-axis the function Ψk has the asymptotic behaviour
Ψk(x, E) = e−λγ
kS(x,E)
q(x, E)−hγ−kh(ψ + o(1)) , x ≫ 0 , (2.9) where ψ is defined as in Theorem 2.4 and γ = e2πih∨. Moreover, let l ∈ Z
+ be
such that 1 ≤ l ≤ h∨
2 . Then, the functions Ψ1−l2 , . . . , Ψ l−1
2 are linearly independent
solutions to equation (2.1) in the representation Vl+1
2 .
Proof. A simple computation shows that S(ωkx, ΩkE) = γkS(x, E), and similarly
q(ωkx, ΩkE) = ωMq(x, E). Therefore by Theorem2.4,
Ψ(ωkx, ΩkE) = ω−Mhe−γkλS(x,E)q(x, E)−h(ψ + o(1)) , x ≫ 0 . Hence, on the positive real semi axis we have
Ψk(x, E) = e−γ
kλS(x,E)
q(x, E)−hγ−kh(ψ + o(1)) , x ≫ 0 , where we have used the identity ωM+1 = γ. Note that γk, k ∈ {1−l
2 , . . . , l−1
2 },
are pairwise distinct. Therefore, the vectors ψk are linearly independent. This
concludes the proof.
Remark 2.8. A slight variation of the proof of Theorem2.4presented below allows one to prove that the asymptotic behavior of Ψ holds on a larger sector of the complex plane, consisting of three adjacent Stokes sectors. This refined result is necessary to consider the lateral connection problem, where one considers the linear
relations among solutions that are subdominant in different Stokes sectors. Such a problem leads naturally to another instance of the ODE/IM correspondence, namely to spectral determinants related to the transfer matrix of integrable systems, see e.g. [28].
2.1. Proof of Theorem2.4. To complete our analysis, we need to further simplify equation (2.5) with a new gauge transformation. Let us denote by eL(x, E) = q(x, E)ad hL(x, E), where L(x, E) and h are as in Lemma2.1.
Lemma 2.9. Let ℓ = Pi∈Iℓihi and h = Pi∈Iaihi, with ℓi, ai ∈ C, and
con-sider the element N = Pi∈I(ℓi− M ai)fi ∈ g . Then we have the following gauge
transformation:
eα(x) ad NL(x, E) = ∂ + q(x, E) Λ + O(xe −1−M) , where α(x) = (x q(x, E))−1 .
Proof. It follows by a straightforward computation using the explicit form of eL(x, E) given in equation (2.4), together with the definitions of N and α(x) and the com-mutation relations (see (1.1))
[fi, e0] = 0 and [fi, ej] = δijhi, for every i, j ∈ I .
By Lemma 2.9 and equation (1.18), it follows that the ODE (2.5) is transformed into
e
Ψ′(x) +q(x, E) Λ + eA(x)Ψ(x) = 0 ,e (2.10)
where eA(x) = O(x−1−M), and eΨ(x) = G(x)Ψ(x), with G(x) = eα(x)N. Since
G(x) =1+ o(x
−M), then we look for a solution to equation (2.10) with the same
asymptotic behavior (2.7) as the solution Ψ(x) to equation (2.5). Finally, we define Φ(x) = eλS(x,E)Ψ(x). Then, it is easy to show that equation (e 2.10) takes the form
Φ′(x) +q(x, E) ˜Λ + eA(x)Φ(x) = 0 , (2.11) where ˜Λ = Λ − λ1. Note that ˜Λ ψ = 0. The problem is now reduced to prove the existence of a solution to equation (2.11) satisfying, for every ε > 0, the asymptotic condition
Φ(x) = ψ + o(1) , in the sector | arg x| < π
2(M + 1)− ε . (2.12) To proceed with the proof we need to introduce some elements of WKB analysis, for which we refer to [16, Chap. 3]. We fix κ > 0 (possibly depending on E) big enough such that the Stokes sector
Σ = {x ∈ eC| Re S(x, E) ≥ Re S(κ, E)}
has the following two properties: S is a bi-holomorphic map from Σ to the right half-plane {z ∈ C | Re z ≥ 0}, and Σ coincides asymptotically with the sector | arg x| < π
2(M+1). In other words, any ray of argument | arg x| < π
2(M+1) eventually
lies inside Σ, while any ray of argument | arg x| > π
2(M+1) eventually lies in the
complement of Σ. For any ε > 0, we define Dε= S−1{| arg z| ≤ π2 − ε}, and we
introduce the space B of holomorphic bounded functions U : Dε→ V for which the
limit limx→∞U (x) = U (∞) is well-defined. The space B is complete with respect
to the norm kU k∞= supx∈Dε|U (x)|.
We construct the solution of equation (2.11) with the desired asymptotic behavior (2.12) as the solution of the following integral equation in the Banach space B:
Φ(x) = K[Φ](x) + ψ(i), (2.13)
where the Volterra integral operator K is defined as
K[Φ](x) = Z ∞ x e− ˜Λ S(x,E)−S(y,E) e A(y)Φ(y)dy .
The integral above is computed along any admissible path c, and we say that a (piecewise differentiable) parametric curve c is admissible ifR0t| ˙c(s)|ds = O(|c(t)|) and Re S(c(t), E) is a non decreasing function. For example, S−1(t+x) is an
admis-sible path [16]. Note that a solution Φ(x) of equation (2.11), with the asymptotic behavior (2.12) belongs to B. Moreover, it satisfies equation (2.10) if and only if it satisfies the Volterra integral equation (2.13). Indeed, provided that K is a continuous operator, we have that limx→∞K[Φ](x) = 0 and
K[Φ]′(x) = − eA(x)Φ(x) − q(x, E) ˜ΛK[Φ](x) = − eA(x)Φ(x) − q(x, E) ˜ΛΦ(x) − ψ(i)
= − eA(x)Φ(x) − q(x, E) ˜ΛΦ(x) ,
where the last equality follows from the fact that ˜Λψ(i) = 0.
In order to prove that the solution of the integral equation (2.13) exists and it is unique we start showing that K is a continuous operator on B and its spectral radius is zero, i.e. limn→∞kKnk
1
n = 0. By hypothesis, all eigenvalues of Λ have
non-positive real part. Since Λ is semi-simple, i.e. diagonalizable, we deduce that there exists a β > 0 such that for any κ ≥ 0 and any ϕ ∈ V ,
|eκ ˜Λϕ| ≤ β|ϕ| . (2.14)
Since | eA(x)| = O(x−1−M), we have that | eA| ∈ L1(c, | ˙c(t)|dt) for every admissible
path c , and the function
ρc(x) = β
Z ∞ x
| eA(c(s))|| ˙c(s)|ds (2.15)
is well-defined and satisfies limx→∞ρc(x) = 0. Moreover, the inequality (2.14)
implies that Z ∞ x e− ˜Λ S(x,E)−S(y,E) e A(y)U (y)dy ≤ ρc(x)kU k∞,
for every U ∈ B. By (2.14) and (2.15) it follows [16] that K[U ] is path-independent and thus well defined. Furthermore, we have
|K[U ](x)| ≤ kU k∞ρ(x) and kKk = sup |U|∞=1 kK[U ]k∞≤ ¯ρ , where ρ(x) = inf cadmissible ρc(x) , ρ = sup¯ x∈Dε ρ(x) .
Note that ρ(x) is a bounded function such that limx→∞ρ(x) = 0. Hence, we
conclude that K is a well-defined bounded operator on B. By definition, Kn[U ]
can be written in terms of K(x, y) as
where K(x, y) is the matrix-valued function
K(x, y) = e− ˜Λ S(x,E)−S(y,E)
e A(y) , and from this it follows that Kn[U ](x) can be estimated as
|Kn[U ](x)| = kU k∞ Z ∞ x . . . Z ∞ yn−1
| eA(y1)| . . . | eA(yn)|dy1. . . dyn ≤ kU k∞ρ n(x)
n! . Thus
kKnk ≤ ρ¯n
n! , (2.16)
and the series
Φ = X
n∈Z+
Kn[ψ]
converges in B. Clearly Φ satisfies the integral equation (2.13), since
K[Φ] =
∞
X
n=1
Kn[ψ] = Φ − ψ .
Moreover, Φ is the unique solution of equation (2.13), for if ˜Φ is another solution then
Kn[Φ − ˜Φ] = Φ − ˜Φ , for every n ∈ Z +.
By equation (2.16), it follows that Φ = ˜Φ.
It remains to prove that Φ is an entire function of E. This follows, by a stan-dard perturbation theory argument, from the fact that eA and S are holomorphic functions of E in Dε. Theorem2.4is proved.
3. The Ψ-system
In this section, for a simple Lie algebra g of ADE type, we prove an algebraic identity known as Ψ-system, which was conjectured first in [9]. Recall from Section
1.3 that we can write
2 ^ V1(i) 2 ∼ = W(i)⊕ U , M(i)=O j∈I V(j)⊗Bij ∼ = W(i)⊕ eU , (3.1)
where the representation W(i)= L(η
i)p(i)+1
2 , i ∈ I, was defined in equation (1.12),
and U and eU are direct sums of all the irreducible representations different from W(i). Due to (1.11), for every i ∈ I we have a morphism of representations
mi: ^ V1(i) 2 → M (i), Ker m i = U .
The Ψ-system is a set of quadratic relations, realized by means of the morphisms mi, among the subdominant solutions to the linear ODE (2.1) defined on the
rep-resentationsVV1(i) 2 and M
(i).
In order to prove the main result of this section, we need to study the maximal eigenvalue (and the corresponding eigenvector) of Λ in the representationsV2V1(i)
2
, M(i) and W(i). To this aim, we recall some further facts from the theory of simple
Lie algebras and simple Lie groups. First, we need the following
Lemma 3.1. Let h ∈ h satisfy relations (1.14), and let us set γ = e2πih∨. Then, for
every k ∈ C, the following formula holds: γkad hΛ = γkM
−k(Λ) . (3.2)
In particular, if ψ is an eigenvector of Λ in an evaluation representation V , then ψk = γ−khψ is an eigenvector of Λ in the representation Vk, and we have
Λψ = λψ if and only if Λψk= γkλψk. (3.3)
Proof. It follows directly from the commutation relations (1.14). Now consider the element ¯Λ = Λ|t=1∈ g, which is well known [25] to be a regular
semisimple element of g. Therefore, its centralizer gΛ¯ = {a ∈ g | [¯Λ, a] = 0} is a
Cartan subalgebra of g, which we denote h′. Due to Lemma3.1, it follows that
γad hΛ = γ ¯¯ Λ , (3.4)
where h ∈ h is the element introduced in Section 1 satisfying relations (1.14). Equation (3.4) says that ¯Λ is an eigenvector of γad h, and this in turn implies that
h′ is stable under the action of γad h. Since γad his a Coxeter-Killing automorphism
of g [25], we can regard γad has a Coxeter-Killing automorphism of h′. From these
considerations it follows – as proved in [6] – that we can choose a set of Chevalley generators of g, say {f′
i, h′i, e′i | i ∈ I} ⊂ g, such that h′ =
L
i∈ICh′i and that the
identity
21n+ γ
ad h+ γ− ad h= (Bt)2, (3.5)
holds as operators on h′. Here B denotes, as usual, the incidence matrix of g.
Following [18], relation (3.5) can be used to express – in the basis {h′
i, i ∈ I} –
the eigenvectors of γad h in terms of the eigenvectors of B. Now it is well known
that the incidence matrix B is a non-negative irreducible matrix [24]. Therefore, it has a maximal (in the sense of the Perron-Frobenius theory) eigenvalue, which is γ12+ γ−12, and a corresponding eigenvector, say (x1, x2, . . . , xn) ∈ Rn
>0. If we fix
x1= 1, we then have the following relation for every i ∈ I:
X j∈I Bijxj = (γ− 1 2 + γ 1 2)x i, xi> 0 . (3.6)
Following [18], and essentially using (3.5), we finally obtain ¯
Λ =X
i∈I
γp(i)2 xih′
i, (3.7)
where xi, i ∈ I, are defined in (3.6) and the function p(i), i ∈ I, is given in (1.8).
Remark 3.2. Note that equation (3.7) corresponds to a precise normalization of ¯Λ. In fact, it is always possible to find an automorphism of g, under which h is stable, in such a way that ¯Λ = cPi∈Iγ±
p(i)
2 xih′
ifor any c ∈ C∗and any choice of the relative
phase γ±p(i)2 . The choice of the relative phase is however immaterial. Indeed, under
the transformation γp(i)2 → γ− p(i)
2 the right hand side of (3.7) represents a different
element in the Cartan subalgebra h′which is however conjugated to ¯Λ. Hence, their
spectra coincide.
The decomposition (3.7) allows one to compute the spectrum of Λ using the Perron-Frobenius eigenvector of the incidence matrix B. We will also need the following:
Lemma 3.3. Let αj, j ∈ I, be the positive roots of g with respect to the Cartan
subalgebra h′. Then ¯Λ satisfies
αj(γ− p(i) 2 Λ) = x¯ jγ p(j)−p(i) 2 (1 − γ−2p(j)+1) , for all i, j ∈ I , (3.8)
where the xi are defined in (3.6). This implies that
Proof. Since Bij6= 0 if and only if p(i) 6= p(j), equation (3.6) implies that X j∈I γ−p(j)2 B ijxj= (γ− 1 2 + γ 1 2)γ p(i)−1 2 x i. (3.9)
Equation (3.9) and the decomposition (3.7) imply the thesis. Using the above lemma, together with equation (3.7), we compute the maximal eigenvalue of Λ in the representations V(i), VV(i), W(i) and M(i).
Proposition 3.4. Let i ∈ I. If g is a simple Lie algebra of ADE type, then the following facts hold.
(a) For the fundamental representation V(i) there exists a unique maximal
eigen-value λ(i) of Λ. The eigenvalues λ(i), i ∈ I, satisfy the following identity:
X j∈I Bijλ(j)= (γ− 1 2 + γ 1 2)λ(i). (3.10)
We denote by ψ(i)∈ V(i) the unique (up to a constant factor) eigenvector of Λ
corresponding to the eigenvalue λ(i) .
(b) For the representation M(i), we have thatP
j∈IBijλ(i)is a maximal eigenvalue
of Λ. Moreover,
ψ(i)⊗ =
O
j∈I
ψ(j)⊗Bij ∈ M(i) is the corresponding eigenvector.
(c) For the representationV2V1(i) 2
, we have that (γ−1
2+γ
1
2)λ(i)is a maximal
eigen-value of Λ. Moreover, γ−12hψ(i)∧ γ 1 2hψ(i)∈ 2 ^ V1(i) 2
is the corresponding eigenvector. (d) The elements γ−1
2hψ(i)∧ γ12hψ(i) ∈V2V1(i)
2 and ψ
(i)
⊗ ∈ M(i) belong to the
sub-representation W(i). Hence, for the representation W(i) there exists a maximal
eigenvalue µ(i) of Λ, given by
µ(i) =X j∈I Bijλ(i)= (γ− 1 2 + γ 1 2)λ(i),
and the relation
mi γ− 1 2hψ(i)∧ γ 1 2hψ(i)= cψ(i) ⊗
holds for some c ∈ C∗.
Proof. Recall that the weights appearing in the representation L(ωi) are of the
following type:
ωi, ωi− αi, ωi− αi− αj− ω ,
where in the latter case the index j is such that Bij 6= 0 and ω is an integral
non-negative linear combination of the positive roots. Clearly for any ˜h ∈ h′ such that
Re αj(˜h) ≥ 0, ∀j ∈ I we have
Re ωi(˜h) ≥ (ωi− αi)(˜h) ≥ (ωi− αi− αj− ω)(˜h) .
Therefore for any such ˜h the eigenvalues with maximal real part is ωi(˜h), followed
by (ωi− αi)(˜h), (ωi− αi− αj)(˜h), and so on.
We use this simple argument to study the maximal eigenvalues of Λ in the representation V(i). By definition of Λ and ¯Λ, and using equation (3.2), the action
of Λ on V(i) coincides with the action of γp(i)2 (−1+ad h)Λ on L(ω¯
αj(γ−
p(i)
2 Λ) ≥ 0, for every j ∈ I, and α¯ i(γ− p(i)
2 Λ) = x¯ i(1 − γ1−2p(i)). It follows
that in the representation V(i), Λ has the maximal eigenvalue
ωi(γ−
p(i)
2 Λ) = x¯ i.
Therefore, by definition of the xi’s, the λ(i) satisfy equation (3.6).
Moreover we have
ψ(i)= γp(i)2 hv′
i (3.11)
and
fi′ψ(i) = cγ−hψ(i), (3.12)
for some c ∈ C∗. This completes the proof of part (a). Part (b) follows directly from
part (a). To prove part (c), we follow the same lines we used to prove part (a). In fact, the action of Λ on V1(i)
2 coincides with the action of γ p∗ (i)
2 (−1+ad h)Λ on L(ω¯ i),
where p∗(i) = 1 − p(i). By equation (3.8) we have Re α i(γ
p∗ (i)
2 (−1+ad h)Λ) = 0,¯
while Re αj(γ
p∗ (i)
2 (−1+ad h)Λ) > 0 if p(i) 6= p(j). It follows that the two eigenvalues¯
with maximal real part correspond to to the weights ωi and ωi− αi. By equation
(3.8), these eigenvalues are xiγ
1
2 and xiγ− 1
2, and the corresponding eigenvectors
are γ∓1
2hψ(i). Therefore VV(i) has a unique maximal eigenvalue (γ12 + γ−12)xi,
with eigenvector γ−12hψ(i)∧ γ12hψ(i) = cγ (1+p(i)) 2 h(v′ i∧ f ′ iv ′ i) , (3.13)
for some c 6= 0. Here we have used equations (3.11) and (3.12).
Let us prove (d). By equation (3.11) and Lemma1.2, it follows that ψ(i)⊗ ∈ W(i),
where W(i)is seen as a subrepresentation of M(i). Therefore the maximal eigenvalue
of W(i) and M(i) coincide. Finally by equations (3.11), (3.13) and Lemma1.1, we
have that γ−1
2hψ(i)∧ γ12hψ(i) ∈ W(i) and m
i(γ−
1
2hψ(i)∧ γ12hψ(i)) = cψ(i)
⊗ , c ∈ C∗,
thus proving part (d) and concluding the proof.
Remark 3.5. We note that we can always choose a normalization of the eigenvectors ψ(i)∈ V(i), i ∈ I, such that
mi γ− 1 2hψ(i)∧ γ 1 2hψ(i)= ψ(i) ⊗ , (3.14)
for every i ∈ I. From now on, we will assume that Proposition 3.4(d) holds with the normalization constant c = 1.
By Proposition3.4(a), for any fundamental representation V(i), i ∈ I ofbg, there
exists a maximal eigenvalue λ(i) with eigenvector ψ(i). Therefore, by Theorem2.4
there exists a unique subdominant solution Ψ(i)(x, E) : eC→ V(i) of the differential
equation (2.1) in the representation V(i) with asymptotic behavior
Ψ(i)(x, E) = e−λ(i)S(x,E)q(x, E)−h ψ(i)+ o(1), in the sector | arg x| < π 2(M + 1).
(3.15) Using Proposition3.4we can prove the following theorem establishing the Ψ-system associated to the Lie algebra g.
Theorem 3.6. Let g be a simple Lie algebra of ADE type, and let the solutions Ψ(i)(x, E) : eC → V(i), i ∈ I, have the asymptotic behavior (3.15). Then, the
following identity holds: mi Ψ(i)−1
2
(x, E) ∧ Ψ(i)1 2
Proof. Due to Proposition 3.4(b) and Theorem2.4, the unique subdominant solu-tion to equasolu-tion (2.1) in M(i) is
⊗j∈IΨ(j)(x, E)⊗Bij = e−µ (i) S(x,E)q(x, E)−hψ(i) ⊗ + o(1) , for x ≫ 0 .
Moreover, by equation (2.8), Corollary2.7and Proposition3.4(c) we have that Ψ(i)−1
2
(x, E) ∧ Ψ(i)1 2
(x, E) = e−µ(i)S(x,E)q(x, E)−hψ−(i)1 2
∧ ψ(i)1 2
+ o(1), for x ≫ 0 . The proof follows by equation (3.14) and the uniqueness of the subdominant
solu-tion.
Example 3.7. In type An, equation (3.16) becomes
mi Ψ (i) −1 2 ∧ Ψ (i) 1 2
= Ψ(i−1)⊗ Ψ(i+1), for i = 1, . . . , n ,
where we set Ψ(0)= Ψ(n+1)= 1.
Example 3.8. In type Dn, equation (3.16) becomes
mi Ψ (i) −1 2 ∧ Ψ (i) 1 2
= Ψ(i−1)⊗ Ψ(i+1), for i = 1, . . . , n − 3 , (3.17)
mn−2 Ψ(n−2)−1 2 ∧ Ψ (n−2) 1 2 = Ψ(n−3)⊗ Ψ(n−1)⊗ Ψ(n), (3.18) mn−1 Ψ(n−1)−1 2 ∧ Ψ(n−1)1 2 = Ψ(n−2)= m n Ψ(n)−1 2 ∧ Ψ(n)1 2 , (3.19) where we set Ψ(0)= 1.
Remark 3.9. The Ψ-system in Example 3.8 is a complete set of relations among the subdominant solutions. The analogue system of equations obtained in [32, Eqs. (3.19,3.20,3.21)] lacks of the equation (3.18) for the node n − 2 of the Dynkin diagram. Note that equations [32, Eqs. (3.19,3.20)] are equivalent to (3.17) and (3.19) of the present paper. Indeed, it follows from the direct sum decomposition3.1
that mi◦ ι =1
V(n−2), for i = n − 1, n, where ι is the embedding V(n−2)
ι
֒→V2V1(i) 2
(in fact, V(n−2) ∼= W(n−1)∼= W(n)in this particular case). The further identity [32,
Eq. (3.21)], which can be obtained from (A.4), requires the introduction of another unknown and therefore equations [32, Eqs. (3.19,3.20,3.21)] are an incomplete system of n equations in n + 1 unknowns.
Example 3.10. In type En, n = 6, 7, 8, equation (3.16) becomes
mi Ψ(i)−1 2
∧ Ψ(i)1 2
= Ψ(i−1)⊗ Ψ(i+1), for i = 1, . . . , n − 4 ,
mn−3 Ψ(n−3)−1 2 ∧ Ψ (n−3) 1 2 = Ψ(n−4)⊗ Ψ(n−2)⊗ Ψ(n−1), mn−2 Ψ(n−2)−1 2 ∧ Ψ (n−2) 1 2 = Ψ(n−3), mn−1 Ψ(n−1)−1 2 ∧ Ψ(n−1)1 2 = Ψ(n−3)⊗ Ψ(n), mn Ψ(n)−1 2 ∧ Ψ (n) 1 2 = Ψ(n−1), where we set Ψ(0)= 1. 4. The Q-system
In this section, for any simple Lie algebra of ADE type, we define the generalized spectral determinants Q(i)(E; ℓ), i ∈ I, and we prove that they satisfy the Bethe
Ansatz equations, also known as Q-system. The spectral determinants are entire functions of the parameter E, and they are defined by the behavior of the functions
Ψ(i)(x, E) close to the Fuchsian singularity. More precisely, Q(i)(E; ℓ) is the
coeffi-cient of the most singular term of the expansion of Ψ(i)(x, E) in the neighborhood
of x = 0.
In the case of the Lie algebra sl2, the spectral determinant Q(E; ℓ) was originally
introduced in [14,3], while the generalization to sln has been given in [10,34]. An
incomplete construction for Lie algebras of BCD type can be found in [9,32]. The terminology generalized spectral determinants is motivated by the sl2case. Indeed,
for the Lie algebra sl2, the linear ODE (2.1) is equivalent to a Schrödinger equation
with a polynomial potential and a centrifugal term, and the spectral determinant Q(E; ℓ) vanishes at the eigenvalues of the Schrödinger operator.
Before proving the main result of this section, we briefly review some well-known fundamental results from the theory of Fuchsian singularities. Here we follow [22, Chap. III].
Remark 4.1. In this Section we assume that the potential p(x, E) = xM h∨
is ana-lytic in x = 0, namely M h∨∈ Z
+. With this assumption, the point x = 0 is simply
a Fuchsian singularity of the equation. The case of a potential with a branch point at x = 0 can formally be treated without any modification – see [3] for the case A1
– but the mathematical theory is less developed.
4.1. Monodromy about the Fuchsian singularity. For any linear ODE with Fuchsian singularity at x = 0, namely of the type
Ψ′(x) = A
x + B(x)
Ψ(x) , (4.1)
where A is a constant matrix and B(x) is a regular function, the monodromy operator is the endomorphism on the space of solutions of (4.1) which associates to any solution its analytic continuation around a small Jordan curve encircling x = 0. More concretely, if we fix a matrix solution Y (x) of the linear ODE (4.1) with non-vanishing determinant (i.e. a basis of solutions), then the monodromy matrix M is defined by the relation
Y (e2πix) = Y (x)M .
Here, Y (e2πix) is the customary notation for the analytic continuation of Y (x) .
We are interested in the invariant subspaces of the monodromy matrix M . By the general theory, the algebraic eigenvalues of the monodromy matrix M are of the form e2πia, for any eigenvalue a of A.
We said that an eigenvalue a of A is non-resonant if there exists no other eigen-value of a′ such that a − a′∈ Z. In this case the eigenvalue e2πia has multiplicity
one and there exists a solution of equation (4.1) which is an eigenvector of the monodromy matrix M with eigenvalue e2πia. Such a solution has the form
χa(x) = xa χa+ O(x)
(4.2) for any χa eigenvector of A with eigenvalue a. If the eigenvalues a1, . . . , ak are
reso-nant, meaning that ai− aj∈ Z for i, j = 1 . . . , k, then the eigenvalue e2πia1 = · · · =
e2πiak has multiplicity k. The matrix M , in general, is not diagonalizable when
restricted to the k-dimensional invariant subspace corresponding to this eigenvalue. The corresponding solutions do not have the form (4.2) due to the appearance of logarithmic terms.
We are interested in the linear ODE (2.1), which has a Fuchsian singularity with A = −ℓ, ℓ ∈ h. For any representation V(i), i ∈ I, the eigenvalues for the action
of ℓ on V(i) have the form λ(ℓ), where λ ∈ P is a weight of the representation
V(i). Note that for a generic choice of ℓ ∈ h and for distinct weights λ
1and λ2, the
eigenvalues are resonant if and only if they coincide, and therefore they correspond to a weight of multiplicity bigger than one.
4.2. The dominant term at x = 0. Let ω ∈ P+, and let us denote by
l∗= max λ∈Pω
Re λ(ℓ) .
Let also λ∗ ∈ P be such that l∗ = Re λ∗(ℓ). If λ∗ is a weight of multiplicity one
in L(ω), then – by the discussion in Section 4.1 – the most singular behavior at x = 0 of a solution to equation (2.1) is x−l∗
. In this section, for any irreducible representation L(ω) of g and a generic ℓ ∈ h, we characterize the weight λ∗ ∈ P
maximizing λ(ℓ) ∈ Z. Moreover, we show that it has multiplicity one. These facts will be used to define the generalized spectral determinant Q(i)(E; ℓ).
It is well-known from the general theory of simple Lie algebras (see [19, Appendix D]), that given a generic element ℓ ∈ h, we can decompose the set of the roots R of gin two distinct and complementary sets (of positive and negative roots)
R+ℓ = {α ∈ R | Re α(ℓ) > 0} , R−ℓ = {α ∈ R | Re α(l) < 0} .
Consequently, we can associate to this ℓ ∈ h a Weyl Chamber Wℓ, as well as an
element wℓof the Weyl group (it is the element which maps the original Weyl
cham-ber into Wℓ) and a (possibly new) set of simple roots ∆ℓ = {wℓ(αi) | i ∈ I}. The
weight wℓ(ω) is the highest weight of L(ω) with respect to the new set of simple
roots ∆ℓ, and due to the definition of R−ℓ, the action of any negative simple root
wℓ(−αi), i ∈ I, decreases the value of Re wℓ(ω)(ℓ). Therefore, λ∗ = wℓ(ω) is the
unique weight maximizing the function Re λ(ℓ), λ ∈ Pω. This weight has
multiplic-ity one since it lies in the Weyl orbit of ω.
In the case of a fundamental representation L(ωi), i ∈ I, the weight wℓ(ωi− αi)
belongs to Pωi, and it has multiplicity one. Moreover, it is possible to show that all
the weights in Pωidifferent from wℓ(ωi) are obtained from wℓ(ωi−αi) by a repeated
action of the negative simple roots wℓ(−αi). We conclude that wℓ(ωi− αi) is the
unique weight maximizing Re λ(ℓ) on Pωi \ {wℓ(ωi)}. We have thus proved the
following result:
Proposition 4.2. Let ℓ ∈ h be a generic element, and let wℓthe associated element
of the Weyl group. Then the weight wℓ(ω) ∈ Pω, ω ∈ P+, has multiplicity one and
Re wℓ(λ)(ℓ) > Re λ(ℓ) ,
for any weight λ ∈ Pω, λ 6= wℓ(ω). In the case of a fundamental weight ω = ωi,
i ∈ I, the weight wℓ(ωi− αi) ∈ Pωi has multiplicity one and
Re wℓ(ωi)(ℓ) > Re wℓ(ωi− αi)(ℓ) > Re λ(ℓ) ,
for any λ ∈ Pωi, λ 6= wℓ(ωi), wℓ(ωi− αi).
Let χ(i), ϕ(i) ∈ L(ω
i) be weight vectors corresponding to the the weights wℓ(ωi)
and wℓ(ωi− αi) respectively. As done in Lemmas1.1and1.2it is possible to show
that L(ηi) is a subrepresentation ofV2L(ωi) (respectivelyNL(ωi)⊗Bij) and that
χ(i)∧ ϕ(i) ∈ L(η
i) (respectively ⊗j∈Iχ(j)⊗Bij ∈ L(ηi)). Moreover, χ(i)∧ ϕ(i) and
⊗j∈Iχ(j)⊗Bij are highest weight vectors (with respect to ∆ℓ) of the
subrepresen-tation L(ηi). Hence, we can identify these vectors (up to a constant) using the
morphism mi defined in equation (1.4). From now on we fix the normalization of
χ(i), ϕ(i), i ∈ I, in such a way that
4.3. Decomposition of Ψ(i). From the above discussion, and in particular form
Proposition 4.2 and equation (4.2), we have that for any evaluation representa-tion Vk(i), i ∈ I and k ∈ C, there exist distinguished (and normalized) solutions χ(i)k (x, E) and ϕ(i)k (x, E) of the linear ODE (2.1) such that they have the most singular behavior at x = 0 and they are eigenvectors of the associated monodromy matrix. Explicitly, these solutions have the asymptotic expansion
χ(i)k (x, E) = x−wℓ(ωi)(ℓ) χ(i)+ O(x),
ϕ(i)k (x, E) = x−wℓ(ωi−αi)(ℓ) ϕ(i)+ O(x).
Since the parameter E appears linearly in the ODE (2.1) and the asymptotic be-havior at x = 0 does not depend on E, by the general theory [22] it follows that χ(i)k (x, E; ℓ) and ϕ(i)k (x, E; ℓ) are entire functions of E.
Using equation (1.17) and comparing the asymptotic behaviors we have that ω−khχ(i)k′(ωkx, ΩkE) = ω −kwℓ(ωi)(ℓ+h)χ(i)(x, E) k+k′ ω−khϕ(i)k′(ω kx, ΩkE) = ω−kwℓ(ωi−αi)(ℓ+h)ϕ(i)(x, E) k+k′, (4.4)
for every k, k′ ∈ C. For generic ℓ ∈ h, the eigenvalues −w
ℓ(ωi)(ℓ), and −wℓ(ωi−
αi)(ℓ) are non-resonant and therefore we can unambiguously define two functions
Q(i)(E; ℓ) and eQ(i)(E; ℓ) as follows. We write
Ψ(i)(x, E, ℓ) = Q(i)(E; ℓ)χ(i)(x, E) + eQ(i)(E; ℓ)ϕ(i)(x, E) + v(i)(x, E) , (4.5) where v(i)(x, E) belongs to an invariant subspace of the monodromy matrix
corre-sponding to lower weights. We call Q(i)(E; ℓ) and eQ(i)(E; ℓ) the generalized spectral
determinants of the equation (1.13). Since Ψ(i)(x, E, ℓ), χ(i)(x, E) and ϕ(i)(x, E)
are entire functions of the parameter E, then also Q(i)(E; ℓ) and eQ(i)(E; ℓ) are
entire functions of the parameter E.
Finally, the Ψ-system (3.16) implies some remarkable functional relations among the generalized spectral determinants. Indeed, we have the following result. Theorem 4.3. Let ℓ ∈ h be a generic element. Then, the spectral determinants Q(i)(E; ℓ) and eQ(i)(E; ℓ) are entire functions of E and they satisfy the following
Q eQ-system: Y
j∈I
Q(j)(E; ℓ)Bij = ω−12θiQ(i)(Ω−12E; ℓ) eQ(i)(Ω 1 2E; ℓ)
− ω12θiQ(i)(Ω21E; ℓ) eQ(i)(Ω−12E; ℓ) ,
(4.6)
where θi= wℓ(αi)(ℓ + h).
Proof. We already showed that Q(i)(E; ℓ) and eQ(i)(E; ℓ) are entire functions of the
parameter E. By the definition of Ψ(i)±1
2(x, E; ℓ) given in (2.8) and from equations
(4.5) and (4.4) we have that Ψ(i)±1 2 (x, E; ℓ) = ω±12wℓ(ωi)(ℓ+h)Q(i)(Ω±12E; ℓ)χ(i) 1 2 (x, E) + ω±12wℓ(ωi−αi)(ℓ+h)Qe(i)(Ω±12E; ℓ)ϕ(i)1 2 (x, E) + ˜v (i)(x, E) . (4.7)
Here we used the fact that, since V−(i)1 2 = V1(i) 2 , then χ(i)1 2 (x, E) = χ(i)−1 2 (x, E) and ϕ(i)1 2 (x, E) = ϕ(i)−1 2
(x, E). Note that ˜v(i)(x, E) belongs to an invariant subspace of
the monodromy matrix corresponding to lower weights. Equation (4.6) is then obtained by equating the component in W(i) in the ψ-system (3.16) by means of
equations (4.5), (4.7) and the normalization (4.3).
We now derive the Q-system (Bethe Ansatz) associated to the Lie algebras of ADE type. Let us denote by Ei ∈ C any zero of Q(i)(E; ℓ). Evaluating equation
(4.6) at E = Ω±1
2Ei we get the following relations
Y j∈I Q(j)(Ω12E; ℓ)Bij = −ω12θiQ(i)(ΩE i; ℓ) eQ(i)(Ei; ℓ) , Y j∈I Q(j)(Ω−12E; ℓ)Bij = ω−12θiQ(i)(Ω−1E i; ℓ) eQ(i)(Ei; ℓ) . (4.8)
Assuming that for generic ℓ ∈ h the functions Q(i)(E; ℓ) and eQ(i)(E; ℓ) do not have common zeros, and taking the ratio of the two identities in (4.8), we get, for any zero Ei of Q(i)(E; ℓ), the Q-system
n Y j=1 ΩβjCij Q(j)ΩCij2 E∗ Q(j)Ω−Cij2 E∗ = −1 , (4.9)
where C = (Cij)i,j∈I is the Cartan matrix of the Lie algebra g, and
βj = 1 2M h∨ X i∈I (C−1)ijθi= 1 2M h∨wℓ(ωj)(ℓ + h) , j ∈ I .
Remark 4.4. The construction of the Q eQ system (4.6) works for any affine Kac-Moody algebra. However, formula (4.9) holds only in the ADE case. In order to define the analogue of equation (4.9) for the Cartan matrix of a non-simply laced algebra g, one has to consider, as suggested in [17], a connection with values in the Langlands dual of the affine Lie algebrabg.
4.4. The case ℓ = 0. The case of ℓ = 0 is not generic and thus the Q functions cannot be defined as in Section 4. It is however straightforward to generalize the generic case to ℓ = 0, either directly or as the limit ℓ → 0. Indeed, we can take the limit ℓ → 0 along sequences such that the element of the Weyl group wℓ = w
is fixed. In this way, the limit solutions χ(i)(x, E, ℓ = 0), ϕ(i)(x, E, ℓ = 0) of (2.1)
satisfy the Cauchy problem
χ(i)k (x, E, ℓ = 0) = vw i , ϕ
(i)
k (x, E, ℓ = 0) = fiwvwi . (4.10)
Here vw
i and fiware respectively the highest weight vector of V(i) and the negative
Chevalley generator corresponding to the element w of the Weyl group. Equations (4.10) define uniquely the two solutions. Then the spectral determinants Q(i), ˜Q(i)
are exactly the coefficients of Ψ(i)(x = 0, E) with respect to the basis element
vw
i , fiwvi. Note that this implies a certain freedom in the choice of the spectral
determinants Q(i), ˜Q(i). In fact, the freedom in the choice of Q(i) corresponds to
the elements of the orbit of ωi under the Weyl group action, while the freedom
in the choice of ˜Q(i) corresponds to the elements of the orbit of the ordered pair
(ωi, αi) under the same group action.
For any choice of wℓ= w, the functions Q(i), ˜Q(i)satisfy (4.8) with ℓ = 0, wℓ= w
and the corresponding Bethe Ansatz equation (4.9), provided they do not have common zeros.
5. g-Airy function
We consider in more detail the special case of (2.1) with a linear potential p(x, E) = x and with ℓ = 0, which is particularly interesting because the sub-dominant solutions and the spectral determinants have an integral representation. Since in the sl2case the subdominant solution coincides with the Airy function, we
call the solution obtained in the general case the g-Airy function. The classical Airy
function models the local behavior of the subdominant solution of the Schrödinger equation with a generic potential close to a turning point [16]; we expect the g-Airy function to play a similar role for equation (2.1).
5.1. Q functions. In the special case of (2.1) with a linear potential p(x, E) = x and with ℓ = 0, it is straightforward to compute the Q functions. By the discussion in Section4.4, in order to define the Q functions we need to chose an element w of the Weyl group and thus the distinguished element vw
i of the basis of V(i). The spectral
Q(i)(E) is then defined as the coefficient with respect to vw
i of Ψ(i)(x = 0, E). Due
to the special form of the potential we have
Ψ(i)(x = 0, E) = Ψ(i)(−E, 0) . and thus, using formula (3.15), we get
Q(i)(E) = e−λ(i) h∨
h∨ +1|E| h∨ +1 h∨ Ci 1 + o(1) as E → −∞ , (5.1)
where Ci is the coefficient of the vector |E|−hψ(i) with respect to the basis vector
vw
i . It is important to note that the asymptotic behavior of the functions Q(i) is
expressed via the eigenvalues Λ(i) which coincide with the masses of the (classical)
affine Toda field theory, as we proved in Proposition3.4. This is precisely the same behavior predicted for the vacuum eigenvalue Q of the corresponding Conformal Field Theory [31].
We conclude that the ODE/IM correspondence depends on the relation among the element Λ ∈bg, the Perron-Frobenius eigenvalue of the incidence matrix B and the masses of the affine Toda field theory.
Remark 5.1. Note that formula (5.1) does not coincide with the asymptotic behavior conjectured in [9, Eq 2.6] for a general potential. In fact, J. Suzuki communicated us [33] that the latter formula is conjectured to hold only for potentials such that
M+1
M h∨ < 1, in which case the Hadamard’s product [9, Eq 2.7] converges. Clearly in
case of a linear potential M+1 M h∨ =
h∨
+1 h∨ > 1.
Remark 5.2. A connection between generalized Airy equations and integrable sys-tems appears, among other places, in [23], where the the orbit of the KdV topolog-ical tau–function in the Sato Grassmannian is described. We note that an operator of the form (1.13) – specialized to the case of algebras of type slnand for ℓ = −h/h∨,
M = 1/h∨ – also appears in that paper. It would be interesting to obtain a more
precise relation between these results and those of the present paper.
5.2. Integral Representation. Given an evaluation representation ofbg, we look for solutions of the equation
Ψ′(x) + e + xe 0 Ψ(x) = 0 , (5.2) in the form Ψ(x) = Z c e−xsΦ(s)ds , (5.3)
where Φ(s) is an analytic function and c is some path in the complex plane. Dif-ferentiating and integrating by parts we get
Z c e−xs − s + e + e0 d ds Φ(s)ds + e−xse0Φ(s) c= 0 , (5.4)
where the last term is the evaluation of the integrand at the end points of the path. We are therefore led to the study of the simpler linear equation
− s + e + e0 d
ds
Φ(s) = 0 , (5.5)
which we solve below for two important examples.
5.3. An−1 in the representation V(1). In this case equation (5.2) is already
known in the literature as the n-Airy equation [16]. Using the explicit form of Chevalley generators ei, i = 0, . . . , n − 1, which is provided in AppendixA.1, and
denoting by Φi(s) the component of Φ(s) in the standard basis of Cn, we find the
general solution of (5.5) as
Φ1(s) = κ e
sn+1
n+1 , Φ
i(s) = si−1Φ1(s) , i = 2, . . . , n ,
where κ is an arbitrary complex number. If we choose the path c as the one connecting e−n+1πi ∞ with en+1πi ∞, then the integral formula (5.3) is well-behaved,
and the boundary terms in (5.4) vanish. Thus, the sln-Airy function has the integral
representation Ψj(x) = 1 2πi Z e+ πin+1∞ e−n+1πi ∞ sj−1e−xs+sn+1n+1ds , j = 1, . . . , n . (5.6)
In case n = 2, the latter definition coincides with the definition of the standard Airy function. By the method of steepest descent we get the following asymptotic expansion of Ψ for x ≫ 0: Ψj(x) ∼ r 1 2πnx 2j−1−n 2n e− n n+1x n+1 n , j = 1, . . . , n . (5.7)
Due to Theorem2.4, the sln-Airy function coincides with the fundamental solution
Ψ(1) of the linear ODE (5.2). In fact the asymptotic behavior of the Airy function
coincides with the asymptotic behavior of the function Ψ(1) (for this computation
one needs to use the explicit formula for h which is given in equation (A.1) ). The solutions Ψ(1)k , k ∈ Z, are obtained by integrating (5.3) along the contour obtained rotating c by e2kπin+1.
5.4. Dn in the standard representation. The second example is the Dn-Airy
function in the representation V(1). Using the Chevalley generators e
i, i = 0, . . . , n,
as described in Appendix A.2, and denoting Φj the jth-component of Φ in the
standard basis of C2n, we find that the general solution of (5.5) reads
Φ1= κ s− 1 2es2n −1 2n−1 Φj= sj−1Φ1(s) , j ≤ n − 1 , Φn= sn−1 2 Φ1(s) , Φn+1= s n−1Φ 1(s) , Φn+j = sn+j−2Φ1(s) , j ≤ n − 1 , Φ2n= s2n−2 2 + 1 4s Φ1(s) .
where κ ∈ C is an arbitrary constant. Let c be the path connecting e−2n−1πi ∞ with
e2n−1πi ∞. Then along c the integral formula (5.3) is well-behaved and it thus defines
a solution of (5.2) that we call Dn-Airy function. Moreover, the boundary terms
in (5.4) vanish. As in the case An, the standard method of steepest descent shows
that such solution is exactly the fundamental solution Ψ(1)to equation (5.2). The
solutions Ψ(1)k , k ∈ Z, are obtained by integrating (5.3) along the contour obtained rotating c by e2n−1kπi .
Appendix A. Action of Λ in the fundamental representations In this appendix we give an explicit description of the maximal eigenvalues of Λ in the fundamental representations and of the corresponding eigenvector.
